Difficulty understanding newtonian mechanics with stationary force

I apologize if this seems much too "freshman", and may come across as annoyingly simple to some more trained or educated on this board. However I will ask the question anyways as I have failed to find an answer elsewhere.

Under Newton's Second Law of Motion, f=ma.
thus F = kg (m/s^2)

But when a person pushes against a wall like in the picture below, there is no acceleration, a=0.

The person and the wall have 0 velocity and 0 acceleration.

If a=0 then F=m(0), F=0
How can there be no force here (treating it mathematically according to the formula), when the person is exerting a force upon a body(the wall)?

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And secondly, if an object of a certain mass and velocity, is in motion but not accelerating,
Newtons second law also would calculate that there is no force according to the formula.

Say a 4 kg bicycle moving at a consistent velocity of 2 m/s, that has 0 acceleration.
http://www.theglobalintelligencer.com/images/bicycle.jpg

F=(4kg)(0) = 0
I know the momentum p, would equal p=(4kg)(2m/s) from p=mv.

But there is no momentum without a force being exerted upon the object. Force is coming from the rotation of the tire via the humans legs rotating a gear (translating an general up and down "force" into a rotational "force", giving the bicycle (the object) a momentum.

1.First law: If an object experiences no net force, then its velocity is constant: the object is either at rest (if its velocity is zero), or it moves in a straight line with constant speed (if its velocity is nonzero).[2][3][4]
2.Second law: The acceleration a of a body is parallel and directly proportional to the net force F acting on the body, is in the direction of the net force, and is inversely proportional to the mass m of the body, i.e., F = ma.
3.Third law: When a first body exerts a force F1 on a second body, the second body simultaneously exerts a force F2 = −F1 on the first body. This means that F1 and F2 are equal in magnitude and opposite in direction.

See if you can apply the correct law(s) to the person pushing on the wall, and to the bicycle.

But when a person pushes against a wall like in the picture below, there is no acceleration, a=0.

That's only true if there are other forces that result in the net force on the person and the wall summing up to zero. For a simple case that only considers forces originating from the person, the person pushes backwards on the surface of the earth, and forwards on the wall. The surface of the earth pushes (or pulls) backwards on the wall, and the net sum of forces on the person and the wall is zero.